Everything that art can do to boost this revival of Tennessee Williams's 1959 play has been done. Marianne Elliott's production is first-rate. The cast, led by Kim Cattrall, is as good as any you'll find in a national company. Yet nothing can persuade me that the play is anything more than overheated melodrama all too rarely alleviated by Williams's instinct for comedy.

The plot defies belief. It concerns two lonely outsiders brought together by circumstance. One is Chance Wayne, a would-be actor and professional gigolo, who returns to his native Gulf Coast town in search of the girl, symbolically named Heavenly, who represents his lost innocence.

Chance's companion, and meal-ticket, is an ageing Hollywood star, Alexandra Del Lago, in flight from the apparently disastrous failure of her last movie. But what complicates their visit is that Heavenly's father, a redneck politician named Boss Finley who runs the town, has threatened Chance with castration if he comes anywhere near his daughter. Williams had an instinctive sympathy with the defeated which shows itself in his touching portrait of the movie-star.

The real problem is Chance. Not only is it hard to summon up much sympathy with this narcissistic parasite who lives off Del Lago and publicly humiliates her. He also seems avid for martyrdom in a way that characterises many of Williams's mid-period heroes: even the idea that he would accept emasculation to atone for his guilt and shame is, on any rational level, ludicrous. But not much in the plot makes sense: the last-minute contrivance that rescues Del Lago's career seems to be based on the bizarre premise that film reviews never reach the Gulf Coast.

The best part of the play is the early bedroom scene between the zonked out Del Lago and the self-seeking Chance. It is an encounter that gives the excellent Cattrall an opportunity to show the movie star's multiple contradictions. Cattrall displays a fear of solitude, a whim of iron, a hunger for the consolation of sex and an acidic wit. Informed by Chance that he was always the best-looking guy in town, she crisply asks "How large is this town?"

Above all, Cattrall conveys the desperation of a woman who knows she is the product of an industry where you're only as good as your last movie.

The American actor, Seth Numrich, does all he can with Chance: he looks suitably handsome while conveying the nervous vanity of those who live by their looks. But it's hard to take as a tragic hero a guy who rushes so eagerly to his doom and the scene where he gets wasted in the hotel bar shows Williams's writing at its flabbiest. Williams is better with the subsidiary characters who are vigorously played. Owen Roe catches the rampant power-mania of Boss Finley who is Big Daddy by another name, Lucy Robinson gives a spirited display as his angry mistress and even the minor role of a liberal-minded heckler is given abundant life by Michael Begley.

Elliott also directs with her customary attention to detail. She even lends the play a dream-like quality by staging unscripted encounters between the movie-star and the ethereal Heavenly and actively acknowledges the melodrama by giving the inevitable lightning-storm an all-out intensity. Rae Smith's pillared set meanwhile conveys the pseudo-Palladian grandeur of the American South. But when all is said and done a vast amount of skill has been expended on a play in which the great Williams seems to be writing from memory.

• This article was amended on 13 June 2013 to correct the spelling of Heavenly in the last paragraph. An earlier version misspelled it as Henley.